By combining a Type I estimate, a Type II estimate, and a Type III estimate together one can get estimates of the form <math>MPZ[\varpi,\delta]</math> or <math>MPZ[\varpi',\delta']</math> for <math>\varpi,\delta</math> small enough. Here are the combinations that have been arisen so far in the Polymath8 project:

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By combining a Type I estimate, a Type II estimate, and a Type III estimate together one can get estimates of the form <math>MPZ[\varpi,\delta]</math> or <math>MPZ[\varpi',\delta']</math> for <math>\varpi,\delta</math> small enough by using the combinatorial lemma. Here are the combinations that have been arisen so far in the Polymath8 project:

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Revision as of 22:26, 25 June 2013

A key input to Zhang's proof that bounded gaps occur infinitely often is a distribution result on primes in smooth moduli, which we have called [math]MPZ[\varpi,\delta][/math] (and later strengthened to [math]MPZ'[\varpi,\delta][/math]. These estimates are obtained as a combination of three other estimates, which we will call [math]Type_I[\varpi,\delta,\sigma][/math], [math]Type_{II}[\varpi,\delta][/math], and [math]Type_{III}[\varpi,\delta,\sigma][/math].

Definitions

Asymptotic notation

[math]x[/math] is a parameter going off to infinity, and all quantities may depend on [math]x[/math] unless explicitly declared to be "fixed". The asymptotic notation [math]O(), o(), \ll[/math] is then defined relative to this parameter. A quantity [math]q[/math] is said to be of polynomial size if one has [math]q = O(x^{O(1)})[/math], and bounded if [math]q=O(1)[/math]. We also write [math]X \lessapprox Y[/math] for [math]X \ll x^{o(1)} Y[/math], and [math]\displaystyle X \sim Y[/math] for [math]X \ll Y \ll X[/math].

If [math]\alpha[/math] is a coefficient sequence and [math]a\ (q) = a \hbox{ mod } q[/math] is a primitive residue class, the (signed) discrepancy[math]\Delta(\alpha; a\ (q))[/math] of [math]\alpha[/math] in the sequence is defined to be the quantity

A coefficient sequence [math]\alpha[/math] is said to be at scale [math]N[/math] for some [math]N \geq 1[/math] if it is supported on an interval of the form [math][(1-O(\log^{-A_0} x)) N, (1+O(\log^{-A_0} x)) N][/math].

A coefficient sequence [math]\alpha[/math] at scale [math]N[/math] is said to obey the Siegel-Walfisz theorem if one has

for any [math]q,r \geq 1[/math], any fixed [math]A[/math], and any primitive residue class [math]a\ (r)[/math].

A coefficient sequence [math]\alpha[/math] at scale [math]N[/math] is said to be smooth if it takes the form [math]\alpha(n) = \psi(n/N)[/math] for some smooth function [math]\psi: {\mathbf R} \rightarrow {\mathbf C}[/math] supported on [math][1-O(\log^{-A_0} x), 1+O(\log^{-A_0} x)][/math] obeying the derivative bounds

[math]\displaystyle \psi^{(j)}(t) = O( \log^{j A_0} x ) [/math]

for all fixed [math]j \geq 0[/math] (note that the implied constant in the [math]O()[/math] notation may depend on [math]j[/math]).

for any fixed [math]C \gt 1[/math] and any congruence class [math]a\ (r)[/math] with [math]r \in {\mathcal S}_I[/math]. Here [math]\tau[/math] is the divisor function.

Smooth and densely divisible numbers

A natural number [math]n[/math] is said to be [math]y[/math]-smooth if all of its prime factors are less than or equal to [math]y[/math]. We say that [math]n[/math] is [math]y[/math]-densely divisible if, for every [math]1 \leq R \leq n[/math], one can find a factor of [math]n[/math] in the interval [math][y^{-1} R, R][/math]. Note that [math]y[/math]-smooth numbers are automatically [math]y[/math]-densely divisible, but the converse is not true in general.

Type I, Type II, and Type III

We say that [math]Type_I[\varpi,\delta,\sigma][/math] holds if, whenever [math] M,N[/math] are quantities with

[math]\displaystyle MN \sim x [/math]

and

[math]\displaystyle x^{1/2-\sigma} \ll N \ll x^{1/2-2\varpi-c}[/math]

or equivalently

[math]\displaystyle x^{1/2+2\varpi+c} \ll M \ll x^{1/2+\sigma}[/math]

for some fixed [math]c\gt0[/math], and [math]\alpha,\beta[/math] are coefficient sequences at scale [math]M,N[/math] respectively with [math]\beta[/math] obeying a Siegel-Walfisz theorem, [math]I \subset [1,x^\delta][/math], and [math](\{a_q\})_{q \in {\mathcal S}_I}[/math] is a congruence class system of controlled multiplicity, then one has

We say that [math]Type_{II}[\varpi,\delta][/math] holds if, whenever [math] M,N[/math] are quantities with

[math]\displaystyle MN \sim x [/math]

and

[math]\displaystyle x^{1/2-2\varpi-c} \ll N \ll x^{1/2}[/math]

or equivalently

[math]\displaystyle x^{1/2} \ll M \ll x^{1/2+2\varpi+c}[/math]

for some sufficiently small fixed [math]c\gt0[/math], and [math]\alpha,\beta[/math] are coefficient sequences at scale [math]M,N[/math] respectively with [math]\beta[/math] obeying a Siegel-Walfisz theorem, [math]I \subset [1,x^\delta][/math], and [math](\{a_q\})_{q \in {\mathcal S}_I}[/math] is a congruence class system of controlled multiplicity, then one has

[math]\alpha,\psi_1,\psi_2,\psi_3[/math] are coefficient sequences at scale [math]M,N_1,N_2,N_3[/math] respectively with [math]\psi_1,\psi_2,\psi_3[/math] smooth, [math]I \subset [1,x^\delta][/math], and [math](\{a_q\})_{q \in {\mathcal S}_I}[/math] is a congruence class system of controlled multiplicity, then one has

There should also be a second "double-primed" variant [math]Type''_I[\varpi,\delta,\sigma], Type''_{II}[\varpi,\delta], Type''_{III}[\varpi,\delta,\sigma][/math] of these estimates, intermediate in strength between the primed and unprimed estimates, in which one assumes a suitable "double dense divisibility" hypothesis, which has not yet been determined precisely.

Note: thus far in the Type III analysis, the controlled multiplicity hypothesis has yet to be used.

The combinatorial lemma

If [math]Type_I[\varpi,\delta,\sigma][/math], [math]Type_{II}[\varpi,\delta][/math], and [math]Type_{III}[\varpi,\delta,\sigma][/math] all hold, then [math]MPZ[\varpi,\delta][/math] holds.

Similarly, if [math]Type'_I[\varpi,\delta,\sigma][/math], [math]Type'_{II}[\varpi,\delta][/math], and [math]Type'_{III}[\varpi,\delta,\sigma][/math] all hold, then [math]MPZ'[\varpi,\delta][/math] holds.

This lemma is (somewhat implicitly) proven here. It reduces the verification of [math]MPZ[\varpi,\delta][/math] and [math]MPZ'[\varpi,\delta][/math] to a comparison of the best available Type I, Type II, and Type III estimates, as well as the constraint [math]\sigma \gt 1/10[/math].

Level 1

This result is implicitly proven here. (There, only [math]Type_I[\varpi,\delta,\sigma][/math] is proven, but the method extends without difficulty to [math]Type'_I[\varpi,\delta,\sigma][/math].) It uses the method of Zhang, and is ultimately based on exponential sums for incomplete Kloosterman sums on smooth moduli obtained via completion of sums.

Level 1

This estimate is implicitly proven here. (There, only [math]Type_I[\varpi,\delta,\sigma][/math] is proven, but the method extends without difficulty to [math]Type'_I[\varpi,\delta,\sigma][/math].) It uses the method of Zhang, and is ultimately based on exponential sums for incomplete Kloosterman sums on smooth moduli obtained via completion of sums.

This estimate is implicitly proven here. (There, only [math]Type_{III}[\varpi,\delta,\sigma][/math] is proven, but the method extends without difficulty to [math]Type'_{III}[\varpi,\delta,\sigma][/math].) It uses the method of Zhang, using Weyl differencing and not exploiting the averaging in the [math]\alpha[/math] or [math]q[/math] parameters. The constraint can also be written as a lower bound on [math]\sigma[/math]:

This estimate is implicitly proven here. It is a refinement of the Level 1 estimate that takes advantage of the [math]\alpha[/math] averaging. The constraint may also be written as a lower bound on [math]\sigma[/math]:

Level 4

It should be possible to improve upon the Level 3 estimate by exploiting averaging in the [math]\alpha[/math] parameter (this was suggested already by Fouvry, Kowalski, Michel, and Nelson).

Combinations

By combining a Type I estimate, a Type II estimate, and a Type III estimate together one can get estimates of the form [math]MPZ[\varpi,\delta][/math] or [math]MPZ[\varpi',\delta'][/math] for [math]\varpi,\delta[/math] small enough by using the combinatorial lemma. Here are the combinations that have been arisen so far in the Polymath8 project: